Maps between K-groups induced by rings homomorphism

Question

Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free abelian group on all isomorphism class of finitely generated modules over $R$ by the subgroup generated by relations coming from short exact sequences.

If $fd_RS<\infty$, one can define a map $f^*:G_0(R)\to G_0(S)$ by: $$f^*([M]) = \sum_{i\geq0} (-1)^i [Tor_R^i(M,S)] $$

(for reference, see Section 7, Chapter 2 of Weibel's book on K-theory.

Now, if $R$ is not regular or $S$ is not a complete intersection in $R$, then having finite flat dimension is somewhat a miraculous condition. So my question is: Can a map $f^*$ be defined in a more general situation than for finite flat dimension maps?

EDIT: Let me elaborate a little bit because of some interesting answers and comments below (especially Clark's answer). The main motivation I have in mind is the case of $R$ being a hypersurface. Then most $R$- modules have infinite resolutions, but it is well known that their resolution is eventually periodic. So, even though they are not homologically finite, the modules can be homologically described with finite data (i.e. finite number of matrices).

The fact above is crucial in many results I know about hypersurfaces. For a random recent example, see here. In particular, in this situation one can define( at least when $S$ is finite $R$-module):

$$f^*([M])= [Tor^{2n}(M,S)] - [Tor^{2n-1}(M,S)]$$

for sufficiently big $n$.

So that's one of the reasons I wonder if there is more systematic map one can define.

Answer 1

Probably not for $G$-theory, unfortunately. That $G_0$ is a contravariant for morphisms of schemes that are globally of finite Tor-dimension is SGA VI, Exp. IV, 2.12. The corresponding statement for the $G$-theory spectra is in Thomason-Trobaugh, 3.14.1.

In effect, given a morphism of schemes $f:X\to Y$, one wishes to show that the induced functor $f^{\star}$ on categories of modules gives rise to a functor between the ∞-categories of cohomologically bounded psuedocoherent complexes of $\mathcal{O}$-modules. Preservation of pseudocoherence is automatic, but the statement that $f^{\star}E$ is cohomologially bounded when $E$ is so is equivalent to the assertion that $f$ is globally of finite Tor-dimension [SGA VI, Exp. III, Pr. 3.3].

This doesn't quite show that there is no way to produce a pullback map on $G$-theory for more general kinds of morphisms, but it does make it clear that it cannot be induced by the functoriality of the ∞-categories of complexes of modules.

Answer 2

My initial thought was to work with something like the Grothendieck group or modules, but instead use a version of the Grothendieck group of the derived category (my first search on Mathscinet turned up a paper from 1969 in Bucur), where your relations come from the triangles instead of short exact sequences. Then the map could just be derived tensor with S. I am a bit rusty with the derived category so I am not sure if this would respect the relations, however.